234 FURTHER CALCULATIONS CONCERNING THE [383
a being the velocity of infinitely small disturbances, and this reduces to (9) when 7 = 1. Whether 7 be greater or less than 1, u is positive when p exceeds pa. Similarly if the law of pressure be that expressed in (6),
p+i
Since jB is positive, values of p greater than p0 are here also accompanied by positive values of u.
By definition the momentum of the wave, whose length may be supposed to be limited, is per unit of cross-section
puda), .................................(13)
the integration extending over the whole length of the wave. If we introduce the value ofu given in (11), we get
rtl
(13)=2fioa f|(£) ' -Adz; ..................(14)
7 — 1 J (\p0J poj
and the question to be examined is the sign of (14). For brevity we may write unity in place of p0, and we suppose that the wave is such that its
r mean density is equal to that of the undisturbed fluid, so that ipdx=l,
J
.where I is the length of the wave. If I be divided into n equal parts, then when n is great enough the integral may be represented by the sum
",-••.......(16)
y+l y+1 7+1
pi * + pa * + ... //Oi+/Qa+..A 2
in which all the p's are positive. Now it is a proposition in Algebra that
/p]+p2+...y \ n )
when |- (7 + 1) is negative, or positive and greater than unity; but that the reverse holds when £ (7 + 1) is positive and less than unity. Of course the inequality becomes an equality when all the n quantities are equal. In the present application the sum of the p's is n, and under the adiabatic law (3), 7 and ^ (7+ 1) are positive. Hence (15) is positive or negative according as •|(7-fl) is greater or less than unity, viz.,,according as 7 is greater or less than unity. In either case the momentum represented by (13) is positive, and the conclusion is not limited to the supposition of small disturbances.
In like manner if the law of pressure be that expressed in (6), we get from (12)
0-1P]\il. 'Trans. 1859, p. 146. that (K— 1) R/\ be small, X (equal to 2?r/&) being the wave-length.